0=x^2+5x-1000000

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Solution for 0=x^2+5x-1000000 equation:


Simplifying
0 = x2 + 5x + -1000000

Reorder the terms:
0 = -1000000 + 5x + x2

Solving
0 = -1000000 + 5x + x2

Solving for variable 'x'.

Combine like terms: 0 + 1000000 = 1000000
1000000 + -5x + -1x2 = -1000000 + 5x + x2 + 1000000 + -5x + -1x2

Reorder the terms:
1000000 + -5x + -1x2 = -1000000 + 1000000 + 5x + -5x + x2 + -1x2

Combine like terms: -1000000 + 1000000 = 0
1000000 + -5x + -1x2 = 0 + 5x + -5x + x2 + -1x2
1000000 + -5x + -1x2 = 5x + -5x + x2 + -1x2

Combine like terms: 5x + -5x = 0
1000000 + -5x + -1x2 = 0 + x2 + -1x2
1000000 + -5x + -1x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
1000000 + -5x + -1x2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-1000000 + 5x + x2 = 0

Move the constant term to the right:

Add '1000000' to each side of the equation.
-1000000 + 5x + 1000000 + x2 = 0 + 1000000

Reorder the terms:
-1000000 + 1000000 + 5x + x2 = 0 + 1000000

Combine like terms: -1000000 + 1000000 = 0
0 + 5x + x2 = 0 + 1000000
5x + x2 = 0 + 1000000

Combine like terms: 0 + 1000000 = 1000000
5x + x2 = 1000000

The x term is 5x.  Take half its coefficient (2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
5x + 6.25 + x2 = 1000000 + 6.25

Reorder the terms:
6.25 + 5x + x2 = 1000000 + 6.25

Combine like terms: 1000000 + 6.25 = 1000006.25
6.25 + 5x + x2 = 1000006.25

Factor a perfect square on the left side:
(x + 2.5)(x + 2.5) = 1000006.25

Calculate the square root of the right side: 1000.003124995

Break this problem into two subproblems by setting 
(x + 2.5) equal to 1000.003124995 and -1000.003124995.

Subproblem 1

x + 2.5 = 1000.003124995 Simplifying x + 2.5 = 1000.003124995 Reorder the terms: 2.5 + x = 1000.003124995 Solving 2.5 + x = 1000.003124995 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + x = 1000.003124995 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + x = 1000.003124995 + -2.5 x = 1000.003124995 + -2.5 Combine like terms: 1000.003124995 + -2.5 = 997.503124995 x = 997.503124995 Simplifying x = 997.503124995

Subproblem 2

x + 2.5 = -1000.003124995 Simplifying x + 2.5 = -1000.003124995 Reorder the terms: 2.5 + x = -1000.003124995 Solving 2.5 + x = -1000.003124995 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + x = -1000.003124995 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + x = -1000.003124995 + -2.5 x = -1000.003124995 + -2.5 Combine like terms: -1000.003124995 + -2.5 = -1002.503124995 x = -1002.503124995 Simplifying x = -1002.503124995

Solution

The solution to the problem is based on the solutions from the subproblems. x = {997.503124995, -1002.503124995}

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